Finite Group Databasebeta

← #6475413

Subgroups — #6475413

Order 512.

1942 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic
3#33218C214yesmaximal, complemented
4#30298C212yesmaximal, complemented
5#30296C212yesmaximal, complemented
6#29299C214yesmaximal, complemented
7#29297C212yesmaximal, complemented
8#29296C212yesmaximal, complemented
9#27955C212yesmaximal, complemented
10#27953C212yesmaximal, complemented
11#25818C212yesmaximal, complemented
12#25816C212yesmaximal, complemented
13#21944C214yesmaximal
14C4×#407C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8—21no—
18C23×C42C2211yescharacteristic
19C2×#501C2214yescomplemented
20C2×#407C2212yes—
21C2×C4×M16C2212yes—
22#2857—21no—
23#2857—21no—
24#2857—21no—
25#2857—21no—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2855—21no—
35#2855—21no—
36#2851C2212yes—
37#2851C2212yes—
38#2848—21no—
39#2848—21no—
40C4×C22⋊Q8—21no—
41C4×C22⋊Q8—21no—
42C4×C22⋊Q8—21no—
43C4×C22⋊Q8—21no—
44#2296C2212yes—
45C2×#391—21no—
46C2×#389—21no—
47C2×#389C2212yes—
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50#2280C2212yes—
51C2×#380—21no—
52C2×#380C2212yes—
53#2084C2214yescomplemented
54#2084C2214yescomplemented
55#2083C2214yescomplemented
56#2083C2214yescomplemented
57#2058C2214yescomplemented
58#2058C2214yescomplemented
59#2057C2214yescomplemented
60#2057C2214yescomplemented
61#1993C2214yescomplemented
62#1992C2214yescomplemented
63#1942C2218yescomplemented
64#1941C2218yescomplemented
65#1854C2218yes—
66#1854C2218yes—
67#1853C2214yes—
68#1853C2214yes—
69#1852C2214yes—
70#1852C2214yes—
71#1847C2214yescomplemented
72#1809C2218yes—
73#1809C2218yes—
74#1749C2212yes—
75#1749C2212yes—
76C4×C22⋊C8C2212yes—
77C4×C22⋊C8C2212yes—
78#1728C2212yes—
79C23×Q8—21no—
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C22×M16C2314yes—
84#533—44no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—44no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#125C2312yes—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—48no—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—22no—
115C22×C42—44no—
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#501—48no—
129C2×#107C2318yes—
130C2×#107C2318yes—
131C2×#106C2318yes—
132C2×#106C2318yes—
133#407Q8116yescomplemented
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C4×M16—44no—
137C2×#97C2314yes—
138C2×C4×C8C2314yes—
139#391—21no—
140#391—21no—
141#391—21no—
142#391—21no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391—22no—
147#389—21no—
148#389—21no—
149#389—21no—
150#389—21no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389C2312yes—
158#389C2314yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#389C2314yes—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#385—22no—
168#385—22no—
169#385—22no—
170#385—22no—
171#385—22no—
172#385—22no—
173#384—21no—
174#384—21no—
175#384C2311yescharacteristic
176#384C2312yes—
177#384C2311yescharacteristic
178#384C2312yes—
179#382C2312yes—
180#381—22no—
181#380—21no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380C2312yes—
194#380C2312yes—
195#380C2314yes—
196#380C2312yes—
197#380C2312yes—
198#380C2314yes—
199#379—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C2×#95—22no—
207C2×#95—22no—
208#340C2318yes—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#336C2314yes—
213#336C2314yes—
214C25C2×D411yescharacteristic
215C22×Q8—21no—
216C22×Q8—21no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—44no—
220C22×Q8—44no—
221C22×Q8—44no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×Q818yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×M16—44no—
252C2×M16—44no—
253C2×M16C2×Q8116yescomplemented
254C22×C8C2418yes—
255C4⋊Q8—28no—
256C4⋊Q8—44no—
257C4⋊Q8—42no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261#125—28no—
262C22⋊Q8—44no—
263C22⋊Q8—44no—
264C22⋊Q8—48no—
265C22⋊Q8—48no—
266C22⋊Q8—48no—
267C22⋊Q8—48no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C4×Q8—44no—
273C4×Q8—44no—
274#117—44no—
275#117—44no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—48no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—48no—
350#107—48no—
351#106—48no—
352#106—48no—
353C22⋊C8C2×Q8116yes—
354C22⋊C8C2×Q8116yescomplemented
355#97—28no—
356C4×C8—44no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371C24—22no—
372C24—22no—
373C24—41no—
374C24—41no—
375C24—41no—
376C24—41no—
377C24C4⋊Q814yes—
378C24C22×D411yescharacteristic
379C24C22×D411yescharacteristic
380C24C22×D411yescharacteristic
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—816no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—416no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4C4⋊Q814yes—
456C22×C4C4⋊Q814yes—
457C22×C4C4⋊Q814yes—
458C22×C4AGL(1,ℤ/8)14yes—
459C22×C4#13714yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×Q814yes—
472C22×C4C22×Q814yes—
473C22×C4C22×Q814yes—
474C22×C4C2511yescharacteristic
475M16—416no—
476C2×C8—28no—
477C2×C8—28no—
478C2×C8C22×Q8116yes—
479C4⋊C4—24no—
480C4⋊C4—28no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—44no—
532C42—44no—
533C42H(5,𝔽2)14yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—28no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23C2×C4⋊Q812yes—
558C23C2×C4⋊Q812yes—
559C23C2×C4⋊Q812yes—
560C23C2×AGL(1,ℤ/8)12yes—
561C23C2×#13712yes—
562C23C23×D411yescharacteristic
563Q8—84no—
564Q8—84no—
565Q8—88no—
566Q8—88no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—88no—
626C2×C4—88no—
627C2×C4C2×C4⋊Q812yes—
628C2×C4C2×C4⋊Q812yes—
629C2×C4C2×C4⋊Q812yes—
630C2×C4#53412yes—
631C2×C4#53412yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4C2×#13712yes—
635C2×C4C2×#13712yes—
636C2×C4#57512yes—
637C2×C4#57512yes—
638C2×C4C23×Q812yes—
639C2×C4C2×H(5,𝔽2)12yes—
640C2×C4#58512yes—
641C8—216no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—416no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22#316514yes—
664C22C22×C4⋊Q811yescharacteristic
665C22C2×#53411yescharacteristic
666C22#344911yescharacteristic
667C22C22×AGL(1,ℤ/8)11yescharacteristic
668C22C22×#13711yescharacteristic
669C22C2×#57511yescharacteristic
670C22C2×#57511yescharacteristic
671C4—22no—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—82no—
683C4—84no—
684C4—82no—
685C4—84no—
686C4—82no—
687C4—82no—
688C4#350114yes—
689C2—24no—
690C2—42no—
691C2—42no—
692C2#3501111yescharacteristic
693C2C2×#316511yescharacteristic
694C2#6066311yescharacteristic
695C2#6075011yescharacteristic
696C2#6075111yescharacteristic
697C2#6095111yescharacteristic
698C2#6095211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #33218C2semidirect
4. #30298C2semidirect
5. #30296C2semidirect
6. #29299C2semidirect
7. #29297C2semidirect
8. #29296C2semidirect
9. #27955C2semidirect
10. #27953C2semidirect
11. #25818C2semidirect
12. #25816C2semidirect
19. C2×#501C22semidirect
53. #2084C22semidirect
54. #2084C22semidirect
55. #2083C22semidirect
56. #2083C22semidirect
57. #2058C22semidirect
58. #2058C22semidirect
59. #2057C22semidirect
60. #2057C22semidirect
61. #1993C22semidirect
62. #1992C22semidirect
63. #1942C22semidirect
64. #1941C22semidirect
71. #1847C22semidirect
133. #407Q8semidirect
253. C2×M16C2×Q8semidirect
354. C22⋊C8C2×Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#366252
C2×#28572
#332182
#302982
#302962
#292992
#292972
#292962
#279552
#279532
#258182
#258162
#219442
C4×#4072